You can see what ColorFunction was used for the graphics as follows.
data = RandomReal[1, {4, 4}];
MatrixPlot[data, PlotTheme -> "Scientific",
PlotLegends ->
Placed[BarLegend[Automatic, LegendLayout -> "Column",
LabelStyle -> {FontSize -> 20}], After]] // InputForm
(*Legended[Graphics[Raster[{{{0.4461453382422279, 0.491477006229007,
0.7637054679655062, 1.}, {0.3154254478896804, 0.3694472079594942,
0.65486047117729, 1.}, {0.7569822664902999, 0.7361394855201956,
0.7844867644266112, 1.}, {0.9143712804788171, 0.8600217622903947,
0.7950091556297764, 1.}}, {{0.6337218541194113, 0.6391201311626331,
0.7762460731231854, 1.}, {0.35374427405760145, 0.4143282314629484,
0.7344141311061161, 1.}, {0.4107447677037708, 0.46361290589486254,
0.7613387293820364, 1.}, {0.6601415148287806, 0.6599152784046868,
0.7780123844964757, 1.}}, {{0.5282720496463704, 0.5561196628831526,
0.7691961268350438, 1.}, {0.33466924070697635, 0.39198654969047086,
0.6948124817847957, 1.}, {0.47836178822829944, 0.5168348572796462,
0.7658593291616591, 1.}, {0.6966959644875594, 0.6886876071400427,
0.780456266758513, 1.}}, {{0.8161315485390368, 0.782696405497182,
0.7884412455516316, 1.}, {0.292198, 0.342242, 0.606638, 1.},
{0.5970079564384867, 0.6102222994870824, 0.7737915308130096, 1.},
{0.30371427829734576, 0.35573047051500206, 0.6305469288199474, 1.}}},
{{0, 0}, {4, 4}}, {0, 1}], Axes -> False, Frame -> True,
FrameLabel -> {None, None}, FrameStyle -> Automatic,
FrameTicks -> {{{{3.5, 1}, {2.5, 2}, {1.5, 3}, {0.5, 4}},
{{3.5, 1}, {2.5, 2}, {1.5, 3}, {0.5, 4}}},
{{{0.5, 1}, {1.5, 2}, {2.5, 3}, {3.5, 4}}, {{0.5, 1}, {1.5, 2}, {2.5, 3},
{3.5, 4}}}}, GridLines -> None, GridLinesStyle ->
Directive[GrayLevel[0.5, 0.4]], LabelStyle -> {FontFamily -> "Times"},
Method -> {"AxisPadding" -> Scaled[0.02], "DefaultBoundaryStyle" -> Automatic,
"DefaultPlotStyle" -> Automatic, "DomainPadding" -> Scaled[0.02],
"RangePadding" -> Scaled[0.05]}],
Placed[BarLegend[{Blend[System`PlotThemeDump`$ThemeColorMatrix, #1] & ,
{0.5478712433217057, 1.}}, LabelStyle -> {FontSize -> 20},
LegendLayout -> "Column", Ticks -> {{0.6506512323727816, 0.3},
{0.7410419717964505, 0.48}, {0.8264110034743599, 0.65}, {0.9117800351522696,
0.8200000000000001}, {0.565282200694872, 0.13}, {0.997149066830179, 0.99}},
"PinningPoint" -> 0.5, "SmoothRange" -> False, Charting`TickSide -> Right,
ColorFunctionScaling -> False], After, Identity]]*)
The ColorFunction used by BarLegend is specified in the output
Blend[System`PlotThemeDump`$ThemeColorMatrix, #1] &